4,799 research outputs found
Construction of classical superintegrable systems with higher order integrals of motion from ladder operators
We construct integrals of motion for multidimensional classical systems from
ladder operators of one-dimensional systems. This method can be used to obtain
new systems with higher order integrals. We show how these integrals generate a
polynomial Poisson algebra. We consider a one-dimensional system with third
order ladders operators and found a family of superintegrable systems with
higher order integrals of motion. We obtain also the polynomial algebra
generated by these integrals. We calculate numerically the trajectories and
show that all bounded trajectories are closed.Comment: 10 pages, 4 figures, to appear in j.math.phys
On convergence towards a self-similar solution for a nonlinear wave equation - a case study
We consider the problem of asymptotic stability of a self-similar attractor
for a simple semilinear radial wave equation which arises in the study of the
Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In
the first step we determine the spectrum of linearized perturbations about the
attractor using a method of continued fractions. In the second step we
demonstrate numerically that the resulting eigensystem provides an accurate
description of the dynamics of convergence towards the attractor.Comment: 9 pages, 5 figure
Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability
The existence of a Lagrangian description for the second-order Riccati
equation is analyzed and the results are applied to the study of two different
nonlinear systems both related with the generalized Riccati equation. The
Lagrangians are nonnatural and the forces are not derivable from a potential.
The constant value of a preserved energy function can be used as an
appropriate parameter for characterizing the behaviour of the solutions of
these two systems. In the second part the existence of two--dimensional
versions endowed with superintegrability is proved. The explicit expressions of
the additional integrals are obtained in both cases. Finally it is proved that
the orbits of the second system, that represents a nonlinear oscillator, can be
considered as nonlinear Lissajous figuresComment: 25 pages, 7 figure
Exact Solutions of a (2+1)-Dimensional Nonlinear Klein-Gordon Equation
The purpose of this paper is to present a class of particular solutions of a
C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry
reduction. Using the subgroups of similitude group reduced ordinary
differential equations of second order and their solutions by a singularity
analysis are classified. In particular, it has been shown that whenever they
have the Painlev\'e property, they can be transformed to standard forms by
Moebius transformations of dependent variable and arbitrary smooth
transformations of independent variable whose solutions, depending on the
values of parameters, are expressible in terms of either elementary functions
or Jacobi elliptic functions.Comment: 16 pages, no figures, revised versio
Helical Magnetorotational Instability in Magnetized Taylor-Couette Flow
Hollerbach and Rudiger have reported a new type of magnetorotational
instability (MRI) in magnetized Taylor-Couette flow in the presence of combined
axial and azimuthal magnetic fields. The salient advantage of this "helical''
MRI (HMRI) is that marginal instability occurs at arbitrarily low magnetic
Reynolds and Lundquist numbers, suggesting that HMRI might be easier to realize
than standard MRI (axial field only). We confirm their results, calculate HMRI
growth rates, and show that in the resistive limit, HMRI is a weakly
destabilized inertial oscillation propagating in a unique direction along the
axis. But we report other features of HMRI that make it less attractive for
experiments and for resistive astrophysical disks. Growth rates are small and
require large axial currents. More fundamentally, instability of highly
resistive flow is peculiar to infinitely long or periodic cylinders: finite
cylinders with insulating endcaps are shown to be stable in this limit. Also,
keplerian rotation profiles are stable in the resistive limit regardless of
axial boundary conditions. Nevertheless, the addition of toroidal field lowers
thresholds for instability even in finite cylinders.Comment: 16 pages, 2 figures, 1 table, submitted to PR
Nonsingular solutions of Hitchin's equations for noncompact gauge groups
We consider a general ansatz for solving the 2-dimensional Hitchin's
equations, which arise as dimensional reduction of the 4-dimensional self-dual
Yang-Mills equations, with remarkable integrability properties. We focus on the
case when the gauge group G is given by a real form of SL(2,C). For G=SO(2,1),
the resulting field equations are shown to reduce to either the Liouville,
elliptic sinh-Gordon or elliptic sine-Gordon equations. As opposed to the
compact case, given by G=SU(2), the field equations associated with the
noncompact group SO(2,1) are shown to have smooth real solutions with
nonsingular action densities, which are furthermore localized in some sense. We
conclude by discussing some particular solutions, defined on R^2, S^2 and T^2,
that come out of this ansatz.Comment: 12 pages, 3 figures. To appear in Nonlinearit
Movable algebraic singularities of second-order ordinary differential equations
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a
(generally branched) solution with leading order behaviour proportional to
(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic
at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each
possible leading order term of this form corresponds to a one-parameter family
of solutions represented near z_0 by a Laurent series in fractional powers of
z-z_0. For this class of equations we show that the only movable singularities
that can be reached by analytic continuation along finite-length curves are of
the algebraic type just described. This work generalizes previous results of S.
Shimomura. The only other possible kind of movable singularity that might occur
is an accumulation point of algebraic singularities that can be reached by
analytic continuation along infinitely long paths ending at a finite point in
the complex plane. This behaviour cannot occur for constant coefficient
equations in the class considered. However, an example of R. A. Smith shows
that such singularities do occur in solutions of a simple autonomous
second-order differential equation outside the class we consider here
Constructing Integrable Third Order Systems:The Gambier Approach
We present a systematic construction of integrable third order systems based
on the coupling of an integrable second order equation and a Riccati equation.
This approach is the extension of the Gambier method that led to the equation
that bears his name. Our study is carried through for both continuous and
discrete systems. In both cases the investigation is based on the study of the
singularities of the system (the Painlev\'e method for ODE's and the
singularity confinement method for mappings).Comment: 14 pages, TEX FIL
Solutions for certain classes of Riccati differential equation
We derive some analytic closed-form solutions for a class of Riccati equation
y'(x)-\lambda_0(x)y(x)\pm y^2(x)=\pm s_0(x), where \lambda_0(x), s_0(x) are
C^{\infty}-functions. We show that if \delta_n=\lambda_n
s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}=
\lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1} and
s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1}, n=1,2,..., then The Riccati equation has
a solution given by y(x)=\mp s_{n-1}(x)/\lambda_{n-1}(x). Extension to the
generalized Riccati equation y'(x)+P(x)y(x)+Q(x)y^2(x)=R(x) is also
investigated.Comment: 10 page
Painlev\'e structure of a multi-ion electrodiffusion system
A nonlinear coupled system descriptive of multi-ion electrodiffusion is
investigated and all parameters for which the system admits a single-valued
general solution are isolated. This is achieved \textit{via} a method initiated
by Painleve' with the application of a test due to Kowalevski and Gambier. The
solutions can be obtained explicitly in terms of Painleve' transcendents or
elliptic functions.Comment: 9 p, Latex, to appear, J Phys A FT
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